arXiv subjects

Nguyen Van Tuyen

Publications and source records attributed to Nguyen Van Tuyen.

At least 19 recordsLinked to original sources

arXiv2026

A proximal subgradient algorithm for constrained multiobjective DC-type optimization

In this paper, we consider a class of constrained multiobjective optimization problems, where each objective function can be expressed by adding a possibly nonsmooth nonconvex function and a differentiable function with Lipschitz continuous gradient, then subtracting a weakly convex function. This encompasses multiobjective optimization problems involving difference-of-convex (DC) functions, which are prevalent in various applications due to their ability to model nonconvex problems. We first establish a necessary optimality condition for these problems and then derive sufficient optimality conditions under some structural assumptions, providing a theoretical foundation for algorithm development. Building on these conditions, we propose a proximal subgradient algorithm tailored to the structure of the objectives. Under mild assumptions, the sequence generated by the proposed algorithm is bounded and each of its cluster points is a stationary point. Numerical experiments on multi-sensor sparse recovery illustrate the effectiveness and computational advantages of the proposed algorithm.

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arXiv2026

Second-Order Optimality Conditions for Sparse Differentiable Optimization Problems via Limiting Second-Order Subdifferentials

In this paper, we investigate a class of sparse optimization problems in which both the objective and constraint functions are Fréchet differentiable and possess locally Lipschitz continuous gradient mappings. More precisely, by utilizing the limiting (Mordukhovich) second-order subdifferential of the associated Lagrangian function, we establish new second-order necessary and sufficient optimality conditions for local optimal solutions. The obtained results are derived under mild assumptions and extend several existing results in the literature. In addition, we apply our theoretical developments to sparse multiobjective optimization problems and derive second-order sufficient optimality conditions for efficient solutions. Several examples are also presented to demonstrate the applicability and effectiveness of the proposed results.

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arXiv2026

Hidden convexity of quadratic systems and its application to quadratic programming

In this paper, we present sufficient conditions ensuring that the sum of the image of quadratic functions and the nonnegative orthant is convex. The hidden convexity of the trust-region problem with linear inequality constraints is established under a newly proposed assumption, which is compared with the previous one in [{\it Math. Program. 147, 171--206, 2014}]. We also provide a complete proof of the hidden convexity of a system of two quadratic functions in [{\it J. Glob. Optim. 56, 1045--1072, 2013}]. Furthermore, necessary and sufficient conditions for the S-lemma concerning systems of quadratic inequalities are investigated. Finally, we derive necessary and sufficient global optimality conditions and strong duality results for quadratic programming.

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arXiv2025

Exact penalty functions in optimization with unbounded constraint sets

This paper identifies necessary and sufficient conditions for the exactness of penalty functions in optimization problems whose constraint sets are not necessarily bounded. The case where the data of problems is locally Lipschitz, semi-algebraic or non-degenerate polynomials is studied in detail. The conditions are given in terms of properties of the objective and residual functions of the problems in question. The obtained results generalize and improve some known results in the literature on exact penalty functions.

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arXiv2025

Directional Subdifferentials at Infinity and Its Applications

This paper investigates the behavior of sets and functions at infinity by introducing new concepts, namely directional normal cones at infinity for unbounded sets, along with limiting and singular subdifferentials at infinity in the direction for extended real-valued functions. We develop several calculus rules for these concepts and then apply them to nonsmooth optimization problems. The applications include establishing directional optimality conditions at infinity, analyzing the coercivity, proving the compactness of the global solution set, and examining properties such as weak sharp minima and error bounds at infinity. To demonstrate the effectiveness of the proposed approach, illustrative examples are provided and compared with existing results.

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arXiv2025

On the existence and the stability of solutions in nonconvex vector optimization

The paper is devoted to the existence of weak Pareto solutions and the weak sharp minima at infinity property for a general class of constrained nonconvex vector optimization problems with unbounded constraint set via asymptotic cones and generalized asymptotic functions. Then we show that these conditions are useful for studying the solution stability of nonconvex vector optimization problems with linear perturbation. We also provide some applications for a subclass of robustly quasiconvex vector optimization problems.

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arXiv2025

On second-order Karush--Kuhn--Tucker optimality conditions for $C^{1,1}$ vector optimization problems

This paper focuses on optimality conditions for $C^{1,1}$ vector optimization problems with inequality constraints. By employing the limiting second-order subdifferential and the second-order tangent set, we introduce a new type of second-order constraint qualification in the sense of Abadie. Then we establish some second-order necessary optimality conditions of Karush--Kuhn--Tucker-type for local (weak) efficient solutions of the considered problem. In addition, we provide some sufficient conditions for a local efficient solution of the such problem. The obtained results improve existing ones in the literature.

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arXiv2025

On approximate Pareto solutions in nonsmooth interval-valued multiobjective optimization with data uncertainty in constraints

This paper deals with approximate Pareto solutions of a nonsmooth interval-valued multiobjective optimization problem with data uncertainty in constraints. We first introduce some kinds of approximate Pareto solutions for the robust counterpart (RMP) of the problem in question by considering the lower-upper interval order relation including: (almost, almost regular) $\mathcal{E}$-Pareto solution and (almost, almost regular) $\mathcal{E}$-quasi Pareto solution. By using a scalar penalty function, we obtain a result on the existence of an almost regular $\mathcal{E}$-Pareto solution of (RMP) that satisfies the Karush--Kuhn--Tucker necessary optimality condition up to a given precision. We then establish sufficient conditions and Wolfe-type $\mathcal{E}$-duality relations for approximate Pareto solutions of (RMP) under the assumption of generalized convexity. In addition, we present a dual multiobjective problem to the primal one via the $\mathcal{E}$-interval-valued vector Lagrangian function and examine duality relations.

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arXiv2025

Error Bounds for a Class of Cone-Convex Inclusion Problems

In this paper, we investigate error bounds for cone-convex inclusion problems in finite-dimensional settings of the form $f(x)\in K$, where $K$ is a smooth cone and $f$ is a continuously differentiable and $K$-concave function. We show that local error bounds for the inclusion can be characterized by the Abadie constraint qualification around the reference point. In the case where $f$ is an affine function, we precisely identify the conditions under which the inclusion admits global error bounds. Additionally, we derive some properties of smooth cones, as well as regular cones and strictly convex cones.

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arXiv2024

On second-order optimality conditions for $C^{1,1}$ optimization problems via Lagrangian functions

This paper focuses on optimality conditions for $C^{1,1}$-smooth optimization problems subject to inequality and equality constraints. By employing the concept of limiting (Mordukhovich) second-order subdifferentials to the Lagrangian function associated with the problem, we derive new second-order optimality conditions for the considered problem. Applications for multiobjective optimization problems are studied as well. These results extend and refine existing results in the literature.

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arXiv2024

Weak sharp minima at infinity and solution stability in mathematical programming via asymptotic analysis

We develop sufficient conditions for the existence of the weak sharp minima at infinity property for nonsmooth optimization problems via asymptotic cones and generalized asymptotic functions. Next, we show that these conditions are also useful for studying the solution stability of nonconvex optimization problems under linear perturbations. Finally, we provide applications for a subclass of quasiconvex functions which is stable under linear additivity and includes the convex ones.

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arXiv2024

Approximate optimality conditions and sensitivity analysis in nearly convex optimization

In this paper, approximate optimality conditions and sensitivity analysis in nearly convex optimization are discussed. More precisely, as in the spirit of convex analysis, we introduce the concept of $\varepsilon$-subdifferential for nearly convex functions. Then, we examine some significant properties and rules for the $\varepsilon$-subdifferential. These rules are applied to study optimality conditions as well as sensitivity analysis for parametric nearly convex optimization problems, which are two important topics in optimization theory.

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arXiv2024

Optimality conditions at infinity for nonsmooth minimax programming

This paper is devoted to study of optimality conditions at infinity in nonsmooth minimax programming problems and applications. By means of the limiting subdifferential and normal cone at infinity, we dirive necessary and sufficient optimality conditions of Karush--Kuhn--Tucker type for nonsmooth minimax programming problems with constraint. The obtained results are applied to a nonsmooth vector optimization problem.

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arXiv2023

Coderivatives at infinity of set-valued mappings

In this paper, the concept of coderivatives at infinity of set-valued mappings is introduced. Well-posedness properties at infinity of set-valued mappings as well as Mordukhovich's criterion at infinity are established. Fermat's rule at infinity in set-valued optimization is also provided. The obtained results, which give new information even in the classical cases of smooth single-valued mappings, provide complete characterizations of the properties under consideration in the setting at infinity of set-valued mappings.

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arXiv2023

On error bounds and optimality conditions at infinity

In this paper, we establish sufficient conditions for the existence of error bounds at infinity for lower semicontinuous inequality systems. We also show that the existence of an error bound at infinity of constraint systems plays an important role in deriving necessary optimality conditions at infinity for constrained optimization problems.

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arXiv2022

Optimality conditions and duality relations in nonsmooth fractional interval-valued multiobjective optimization

This paper deals with Pareto solutions of a nonsmooth fractional interval-valued multiobjective optimization. We first introduce four types of Pareto solutions of the considered problem by considering the lower-upper interval order relation and then apply some advanced tools of variational analysis and generalized differentiation to establish necessary optimality conditions for these solutions. Sufficient conditions for Pareto solutions of such a problem are also provided by means of introducing the concepts of (strictly) generalized convex functions defined in terms of the limiting/Mordukhovich subdifferential of locally Lipschitzian functions. Finally, a Mond--Weir type dual model is formulated, and weak, strong and converse-like duality relations are examined.

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arXiv2022

On approximate quasi Pareto solutions in nonsmooth semi-infinite interval-valued vector optimization problems

This paper deals with approximate solutions of a nonsmooth semi-infinite programming with multiple interval-valued objective functions. We first introduce four types of approximate quasi Pareto solutions of the considered problem by considering the lower-upper interval order relation and then apply some advanced tools of variational analysis and generalized differentiation to establish necessary optimality conditions for these approximate solutions. Sufficient conditions for approximate quasi Pareto solutions of such a problem are also provided by means of introducing the concepts of approximate (strictly) pseudo-quasi generalized convex functions defined in terms of the limiting subdifferential of locally Lipschitz functions. Finally, a Mond--Weir type dual model in approximate form is formulated, and weak, strong and converse-like duality relations are proposed.

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arXiv2020

Approximate solutions of interval-valued optimization problems

This paper deals with approximate solutions of an optimization problem with interval-valued objective function. Four types of approximate solution concepts of the problem are proposed by considering the partial ordering $LU$ on the set of all closed and bounded intervals. We show that these solutions exist under very weak conditions. Under suitable constraint qualifications, we derive Karush--Kuhn--Tucker necessary and sufficient optimality conditions for convex interval-valued optimization problems.

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